Theorem prnz 4722

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4707	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4285	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430  ∅c0 4274  {cpr 4570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3432  df-dif 3893  df-un 3895  df-nul 4275  df-sn 4569  df-pr 4571

This theorem is referenced by:  opnz  5421  propssopi  5456  fiint  9230  wilthlem2  27046  upgrbi  29176  wlkvtxiedg  29708  shincli  31448  chincli  31546  constrextdg2lem  33908  spr0nelg  47948  sprvalpwn0  47955